# What does it mean for a set to be bounded?

what does it mean for a set to be bounded??

in the context of the hein-borel theorem

i mean the mathematically rigorous definition

Office_Shredder
Staff Emeritus
Gold Member

S a subset of Rn is bounded if there exists M>0 so that for all x in S, |x|<M

Another equivalent definition is that it has finite diameter, where

$$\mathrm{diam}(S) = \sup_{x, y \in S}(\mathrm{dist}(x, y)).$$

This is applicable to any metric space (though the Heine-Borel theorem is not!).